When solving for the governing equation for pressure (REF ) for the case where in inviscid fluid is injected into a viscous one, we consider the model in polar coordinates as it is simpler to incorporate the far-field boundary condition () via the Dirichlet-to-Neuman map (sec:Farfield) on a circle. However, when the inviscid region surrounds the viscous fluid, we no longer have a far-field boundary condition and, as such, it is more convenient to solve for pressure in Cartesian coordinates, although of course either coordinate system could be used. The discretisation of the spatial derivatives (REF ) is performed as was described in sec:SolvingPressure. That is, we use central finite differences for nodes away from the interfaces, and incorporating a ghost value of \(p\)  for nodes adjacent to the interface. We refer the reader to [1]} and [2]}, [3]} for more details on implementing this stencil in Cartesian coordinates.
